Euclidean Geometry and its Subgeometries
In this monograph, the authors present a modern development of Euclidean geometry from independent axioms, using up-to-date language and providing detailed proofs. The axioms for incidence, betweenness, and plane separation are close to those of Hilbert.
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Euclidean Geometry and its Subgeometries
Edward John Specht (1915–2011) Harold Trainer Jones (1925–1995) Keith G. Calkins Donald H. Rhoads
Euclidean Geometry and its Subgeometries
Edward John Specht Indiana University South Bend South Bend, IN, USA
Harold Trainer Jones Andrews University Berrien Springs, MI, USA
Keith G. Calkins Ferris State University Big Rapids, MI, USA
Donald H. Rhoads Andrews University Berrien Springs, MI, USA
ISBN 978-3-319-23774-9 ISBN 978-3-319-23775-6 (eBook) DOI 10.1007/978-3-319-23775-6 Library of Congress Control Number: 2015952314 Mathematics Subject Classification (2010): 51M05 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
Preface
“At last I said—Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my fathers house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.” –Abraham Lincoln, quoted by Henry Ketcham, in The Life of Abraham Lincoln.
For centuries, the study of Euclidean geometry has been considered an essential part of a literate person’s education, both for the practical knowledge obtained and, more importantly, as an example of a deductive system in which non-obvious conclusions may be drawn from a collection of accepted statements. One of our esteemed colleagues has remarked to us that in view of the influence that Euclidean geometry has had on western civilization, “someone should do it right.” This book is an attempt to do so. Even though Hilbert’s development of Euclidean geometry Foundations of Geometry (1899) [10] has been judged by some as only partially successful, it has set a st
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